Correlation and regression: finding relationships (and why correlation isn't causation)

Does more advertising bring more sales? Do farms with more rainfall harvest more maize? Correlation measures how strongly two numbers move together, and regression draws the best straight line through them so you can estimate one from the other.

Correlation: the coefficient r

The correlation coefficient r runs from −1 to +1:

A rough guide: above 0.7 (or below −0.7) is strong; 0.3 to 0.7 is moderate; below 0.3 is weak.

import statistics as st

ad_spend = [5, 8, 10, 12, 15, 18, 20, 25]          # KSh thousands per week
sales =    [58, 76, 74, 90, 95, 113, 112, 141]    # KSh thousands per week

r = st.correlation(ad_spend, sales)
print("Correlation r =", round(r, 3))

A value near +1 means weeks with more advertising had more sales.

Correlation is not causation

Two things can move together without one causing the other:

To show causation you usually need an experiment (like an A/B test, next lesson) or very careful analysis that controls for other factors.

Simple linear regression: the line of best fit

Regression finds the straight line y = a + b × x that best fits the points:

• b (slope): how much y changes when x goes up by 1
• a (intercept): the value of y when x is 0

import statistics as st

ad_spend = [5, 8, 10, 12, 15, 18, 20, 25]
sales =    [58, 76, 74, 90, 95, 113, 112, 141]

slope, intercept = st.linear_regression(ad_spend, sales)
print(f"sales ≈ {intercept:.1f} + {slope:.2f} × ad_spend")

for spend in [10, 22, 30]:   # careful: 30k is outside the data (5k to 25k)
    print(f"If we spend {spend}k on ads, expected sales ≈ {intercept + slope * spend:.0f}k")

The slope says roughly how many thousand shillings of extra sales came with each extra thousand spent on ads, in this data.

R-squared: how well does the line fit?

R² (r squared) is the share of the variation in y explained by x, from 0 to 1.

import statistics as st

ad_spend = [5, 8, 10, 12, 15, 18, 20, 25]
sales =    [58, 76, 74, 90, 95, 113, 112, 141]
r = st.correlation(ad_spend, sales)
print(f"R² = {r * r:.2f}  ->  about {r * r:.0%} of the variation in sales moves with ad spend")

Be careful with predictions

• Don't extrapolate far: the data covers ad spend from 5k to 25k. Predicting sales at 200k is guesswork; returns usually shrink.
• Look at the scatter plot first: some relationships are curved, and a straight line misleads.
• Outliers can drag the line: check them (lesson 4).
• Other factors matter: holidays, prices and competitors also affect sales.

Summary

• Correlation r (−1 to +1) measures how strongly two numbers move together in a straight line.
• Correlation is not causation: watch for confounders and reverse causation.
• Regression gives a line y = a + b × x; the slope shows the change in y per unit of x.
• R² shows how much variation the line explains; don't extrapolate far beyond your data.